One of the defining characteristics of the situation of a mean reversion illusion is that mean reversion expectations can be implemented without being noticed by the other market participants, for example by synthesized options. Further-more, the fundamental value of the assets in question is hard to evaluate in this situation. This means that finding the point of the start of the illusion is a much more subtle task than finding the disillusion.
In the notation of Figure 3.2 I look for the time t. That is, I search for a segment of a magnitude of years before the crash where mean reversion expec-tations were relatively high. As expecexpec-tations cannot be measured, I use actual mean reversion as proxy. According to the hypothesis this segment should be followed by a segment with slower mean reversion that leads up to the crash.
The Brady-Report locates the beginning of the bull market that led up to the crash in 1982. The contributing factors are described as “continuing deregulation of the financial markets; tax incentives for equity investing; stock retirements arising from mergers, leveraged buyouts and share repurchase programs; and an increasing tendency to include ‘takeover premiums’ in the valuation of a large number of stocks”. The valuation of the U.S. stock market by the end of 1986 is described as high but not unprecedented in terms of price/earnings ratios.
The appreciation from January 1987 through August 1987, however, “challenged historical precedent and fundamental justification” (Brady et al. (1988), p. 9, I-2).
Using this segmentation as a guideline, I estimate model (3.3.2) on the seg-ments 01/02/82–12/30/86 and 01/02/87–10/15/87. That is, I set t = January 2, 1987. I assume that the mean return holds for the total period; the model (3.3.2) is estimated on the 1987-segment with the mean return set fix at the estimate from the period 1982–1986. Figure 3.6 illustrates the estimations. The estimate of the mean reversion speed on the 1982–1986 segment is significant at the one-sided 0.95 significance level. The estimates switch from a higher to a lower value, supporting the hypothesis.
I use a Generalized Likelihood Ratio (GLR) scheme as a changepoint detector (Lai 1995). Let S = {St}t∈{1,...,N } be the considered time series of index prices.
The GLR scheme sets a changepoint at inf
n∈{1,...,N }
max
1≤k≤n sup
θ∈Θ
n
i=k
log fθ(Si|S1, . . . , Si−1) fθ0(Si|S1, . . . , Si−1)
> c
, (3.4.7)
where N is the number of observations and Θ is the open parameter set. fθ is the probability density given the parameter vector θ. θ0 is the parameter vector of the null hypothesis and c is an a priori constant. There is no analytical expression or distribution result for c so that it must be found by simulation methods.
I decomposed the problem (3.4.7) into the following steps. On a baseline segment of the first m points of the series I estimated model (3.3.2). Thereby I obtain the null hypothesis ˆθ0 = (ˆµ0, ˆλ0, ˆσ0). Then I estimated (3.3.2) on every single subseries {S1, . . . , Sj}, j = m + 1, . . . , N. This gave us a series of ˆθj
maximizing the likelihood functions (3.3.6) of the subseries. From this series I computed the probability densities fθˆj(Sj|S1, . . . , Sj−1) for every j = m+1, . . . , N and stored
Zj := logfθˆj(Sj|S1, . . . , Sj−1) fθˆ0(Sj|S1, . . . , Sj−1). From the resulting series{Zj}j∈{m+1,...,N }, the statistics series
ξn= max
m+1≤k≤n
n j=k
Zj, n = m + 1, . . . , N (3.4.8)
Jan82 Jan83 Jan84 Jan85 Jan86 Jan87 Oct87 120
140 160 180 200 220 240 260 280 300 320
S&P500 index
µ = 0.0005 (4.5e−5)
λ = 0.0060*
(0.0034) σ = 0.0089 (0.0003) λ = 0.0004 σ = 0.0103
Figure 3.6: The bull market January 1982 to October 15, 1987 as seen in the S&P500. Using the segmentation of the Brady-Report, I estimate model (3.3.2) on the period January 1982 to December 1986 and January 1987 to October 15, 1987. I assume that the same mean return holds for the complete period, it is estimated at 0.0005 from the 1982–1986 segment. The figures in parentheses are standard errors according to White (1982). The estimate of the mean reversion parameterλ on the period 1982–1986 is significant at the one-sided 0.95 significance level. The estimate on the 1987 segment is much lower than the estimate before.
was calculated. As I search for a single changepoint only, it is interesting to plot the {ξn} series. Figure 3.7 shows the series when the baseline distribution is estimated on the S&P500 observations January 2, 1982, through December 30, 1985. The series is then calculated for the observations January 2, 1986 through October 15, 1987. It can be seen that the estimated parameters move away from the estimated baseline parameters at two distinct speeds as the sample size increases. This is the interpretation of the two trends in the series that can be distinguished. The trend break is at the turn of the years 1986 to 1987. This supports the observation of the Brady-Report.
A simulation gives the significance levels: I generated 1,000 time series ac-cording to model (3.3.2) with the parameters obtained from the estimation of the sample period January 2, 1982 through December 30, 1985 (ˆµ = 0.0005, ˆλ = 0.006, ˆσ = 0.009). This sample consists of 1,012 observations. The sample pe-riod January 2, 1986 through October 15, 1987 for which the detector series ξn
in Figure 3.7 is depicted consists of 454 observations. Therefore, each of the 1000 simulated time series consisted of 1466 observations. On the first 1,012
observa-01/28/86 05/08/86 08/16/86 11/24/86 03/04/87 06/12/87 09/20/87 0
1 2 3 4 5 6 7
detector statistic ξ n
0.90 significance level (2.72) 0.95 significance level (3.33) 0.99 significance level (4.73)
Figure 3.7: Changepoint detector statistic series {ξn} as given by Equation (3.4.8). The baseline parameter vectorθ0was estimated on the segment January 2, 1982 through December 30, 1985. The detector statistics series was then calculated for the observations January 2, 1986 through October 15, 1987. Two distinct trends can be observed in the statistic. This means that the estimated parameters move away from the estimated baseline parameters by two distinct speeds as the sample size increases. The trend break is almost exactly at the turn of the years 1986 to 1987, in line with the periods as given by the Brady-Report. The significance levels were obtained by simulation of the statistic.
tions of each series model (3.3.2) is estimated. Then for each series the detector statistic ξn is calculated for the remaining 454 observations, yielding 454,000 ob-servations of the detector statistic. The significance levels reported in Figure 3.7 are the quantiles of these 454,000 observations.
With only this information in hand, what would have been the estimate on October 16, 1987, of the magnitude of a possible crash? More precise, with the information available on October 16, 1987, the question is: Given that the mean reversion illusion occurred at the beginning of the year 1987, about 200 days ago, and given that the mean reversion disillusion happens today, what will be the distance in the paths that must be corrected? In the notation of Figure 3.2 I now look for the distance in the trajectories that is shaded black, measured at the point immediately before the crash. Let me emphasize that I do not estimate the time of the crash, the disillusion is assumed to happen today for whatever reason.
I simulated model (3.3.2) with the estimated parameters as reported in Figure
3.6. I generated 10,000 paths of a random walk of length 200. Then I evaluated model (3.3.2) with the parameter vectors obtained from the 1982–1986 segment.
The value 246.45 of the S&P500 on January 2, 1987, was set as the starting point. If a mean reversion illusion occurred in January 1987, it lasted for about 200 days up to October 16, 1987. That is, without the illusion the process would have continued for another 200 days under the old regime. The simulation thus gives an estimate of the distribution of the index value Sno illusion(200) on October 16, 1987, without mean reversion illusion. The actual value of the S&P500 at the closing of October 15, 1987, was 298.08. I am hence interested in the sample distribution of the difference log(Sno illusion(200))−log(298.08). This is an estimate of the distribution of the magnitude of the crash.
Table 3.3 (left) shows the sample distribution of the difference log(Sno illusion(200))− log(298.08). There is still a substantial probability for an upward jump as even under the regime with stronger mean reversion there is a number of paths that end up above 298.08 after 200 days. The probability of a crash of minus 20 per-cent or more was more than seven perper-cent. The probability of a correction of minus ten percent or more was more than 40 percent.
To put the somewhat random endpoint of 298.08 into perspective, I evalu-ated model (3.3.2) for 10,000 sample paths under both parameter regimes, that of the 1982–1986 period (Sno illusion) and that of the 1987 period (Sillusion). Table 3.3 (right) shows the sample distribution of the difference log(Sno illusion(200))− log(Sillusion(200)). Even after only 200 days the difference in the mean reversion parameter λ results in substantial distances in the trajectories and thus substan-tial probabilities for large jumps when a mean reversion disillusion happens.
These sample distributions were calculated under the assumption that if the mean reversion illusion had not occurred, the Brownian sample path could have been different from the one that was realized between January 2, 1987, and October 15, 1987. One might argue that the stream of fundamental information that makes up the noise part would have been the same in either case. Under this assumption, I can reconstruct the Brownian sample path between January 2, 1987, and October 15, 1987, from model (3.3.2) by
εˆt= 1 σˆ
µ + ˆˆ λ log ˆϑt+ (1− ˆλ) log St− log St−1
using the parameter estimates from the 1987 segment.
Setting ˆεtback in into the model with the parameters from the 1982–1986 seg-ment, this gives a point estimate for the Sno illusion(200) and thus a point estimate for the magnitude of the crash. In the case of 1987, I have Sno illusion(200) = 273.78 and thus
log(Sno illusion(200))− log(298.08) = −0.085, a correction of minus 8.5 percent.
Table 3.3: The left table shows the sample distribution of the difference log(Sno illusion(200))− log(298.08), the latter value is that of the S&P500 at the close of October 15, 1987. This gives an estimate of the distribution of the magnitude of the crash. The probability of a downward jump of 20 percent or more was more than seven percent. The right table shows the sample distribution of the difference log(Sno illusion(200))− log(Sillusion(200)) when 10,000 Brownian sample paths of length 200 are evaluated under both regimes, that of the 1982–1986 period (Sno illusion) and that of the 1987 period (Sillusion). This shows that the difference in the mean reversion parameter leads to substantial probabilities for large moves when a mean reversion disillusion occurs.
ri P(ri− 0.10 ≤ r < ri) ri P(ri− 0.10 ≤ r < ri)
-0.5 0.0009
-0.4 0.0053
-0.3 0.0029 -0.3 0.0221
-0.2 0.0753 -0.2 0.0751
-0.1 0.3652 -0.1 0.1572
0 0.4332 0 0.2333
0.1 0.1160 0.1 0.2297
0.2 0.0072 0.2 0.1687
0.3 0.0001 0.3 0.0775
0.4 0.0244
0.5 0.0052
0.6 0.0006
It is conspicuous that the estimated magnitude of the mean reversion param-eter λ is much higher after the crash than in the years 1982 to 1986. One reason for this may be that only a part of the mean reversion expectations after the crash depended on mean reversion expectations prior to the crash. A general increase in risk-aversion after the crash may have caused an autonomous increase in mean reversion expectations.